A144755 -id:A144755 - cp's OEIS frontend

A161508 Numbers k such that 2^k-1 has only one primitive prime factor.

2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208

Offset: 1

T. D. Noe, Jun 17 2009

nonn

Also, numbers k such that A086251(k) = 1.
Also, numbers k such that A064078(k) is a prime power.
The corresponding primitive primes are listed in A161509.
The binary expansion of 1/p has period k and this is the only prime with such a period. The binary analog of A007498.
This sequence has many terms in common with A072226. A072226 has the additional term 6; but it does not have terms 18, 20, 21, 54, 147, 342, 602, and 889 (less than 10000).
All known terms that are not in A072226 belong to A333973.

T. D. Noe, Table of n, a(n) for n=1..179
Wikipedia, Unique prime, section Binary unique primes.

Cf. A007498, A064078, A072226, A086251, A144755, A161509, A247071, A333973.

Select[Range[1000], PrimePowerQ[Cyclotomic[ #,2]/GCD[Cyclotomic[ #,2],# ]]&]

is_A161508(n) = my(t=polcyclo(n,2)); isprimepower(t/gcd(t,n)); \\ Charles R Greathouse IV, Nov 17 2014

A161509 The unique primitive prime factor of 2^n-1 for the n in A161508.

Original entry on oeis.org

3, 7, 5, 31, 127, 17, 73, 11, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 241, 2731, 262657, 331, 2147483647, 65537, 599479, 43691, 174763, 61681, 5419, 2796203, 4432676798593, 87211, 15790321, 2305843009213693951, 715827883

Offset: 1

T. D. Noe, Jun 17 2009

nonn

For these primes p, the binary expansion of 1/p has a unique period length. The binary analog of A007615.

Max Alekseyev, Table of n, a(n) for n = 1..179 (terms for n=1..100 from T. D. Noe)

Cf. A144755 (sorted).

Reap[Do[c=Cyclotomic[n,2]; q=c/GCD[c,n]; If[PrimePowerQ[q], Sow[FactorInteger[q][[1,1]]]],{n,100}]][[2,1]]

A306073 Bases in which 3 is a unique-period prime.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 17, 23, 26, 28, 35, 47, 53, 71, 80, 82, 95, 107, 143, 161, 191, 215, 242, 244, 287, 323, 383, 431, 485, 575, 647, 728, 730, 767, 863, 971, 1151, 1295, 1457, 1535, 1727, 1943, 2186, 2188, 2303, 2591, 2915, 3071, 3455, 3887

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 3^t + 1, t >= 1; (b) b = 2^s*3^t - 1, s >= 0, t >= 1.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 3, there are no nontrivial bases, since ord(3,b) <= 2.

			If b = 3^t + 1, t >= 1, then b - 1 only has prime factor 3, so 3 is a unique-period prime in base b.
If b = 2^s*3^t - 1, t >= 1, then the prime factors of b^2 - 1 are 3 and prime factors of b - 1 = 2^s*3^t - 2, 3 is the only new prime factor so 3 is a unique-period prime in base b.

Jianing Song, Table of n, a(n) for n = 1..805
Wikipedia, Unique prime

Cf. A040017 (unique primes in base 10), A144755 (unique primes in base 2).

Bases in which p is a unique prime: A000051 (p=2), this sequence (p=3), A306074 (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).

p = 3;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A306074 Bases in which 5 is a unique-period prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 19, 24, 26, 39, 49, 79, 99, 124, 126, 159, 199, 249, 319, 399, 499, 624, 626, 639, 799, 999, 1249, 1279, 1599, 1999, 2499, 2559, 3124, 3126, 3199, 3999, 4999, 5119, 6249, 6399, 7999, 9999, 10239, 12499, 12799, 15624, 15626, 15999, 19999, 20479

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 5^t + 1, t >= 1; (b) b = 2^s*5^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 7.
For every odd prime p, p is a a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 5, the nontrivial bases are 2, 3, 7.

			1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is a unique-period prime in base 2.
1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is a unique-period prime in base 3.
1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is a unique-period prime in base 7.

Jianing Song, Table of n, a(n) for n = 1..544
Wikipedia, Unique prime

Cf. A040017 (unique-period primes in base 10), A144755 (base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), this sequence (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).

p = 5;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A306075 Bases in which 7 is a unique-period prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 13, 18, 19, 27, 48, 50, 55, 97, 111, 195, 223, 342, 344, 391, 447, 685, 783, 895, 1371, 1567, 1791, 2400, 2402, 2743, 3135, 3583, 4801, 5487, 6271, 7167, 9603, 10975, 12543, 14335, 16806, 16808, 19207, 21951, 25087, 28671, 33613, 38415, 43903, 50175

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 7^t + 1, t >= 1; (b) b = 2^s*7^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 18, 19.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 7, the nontrivial bases are 2, 3, 4, 5, 18, 19.

			1/7 has period length 3 in base 2. Note that 7 is the only prime factor of 2^3 - 1 = 7, so 7 is a unique-period prime in base 2.
1/7 has period length 3 in base 4. Note that 3, 7 are the only prime factors of 4^3 - 1 = 63, but 1/3 has period length 1, so 7 is a unique-period prime in base 4.
1/7 has period length 3 in base 18. Note that 7, 17 are the only prime factors of 18^3 - 1 = 5831, but 1/17 has period length 1, so 7 is a unique-period prime in base 18.
(1/7 has period length 6 in base 3, 5, 19. Similar demonstrations can be found.)

Jianing Song, Table of n, a(n) for n = 1..448
Wikipedia, Unique prime

Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), this sequence (p=7), A306076 (p=11), A306077 (p=13).

p = 7;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A306076 Bases in which 11 is a unique-period prime.

Original entry on oeis.org

2, 3, 10, 12, 21, 43, 87, 120, 122, 175, 241, 351, 483, 703, 967, 1330, 1332, 1407, 1935, 2661, 2815, 3871, 5323, 5631, 7743, 10647, 11263, 14640, 14642, 15487, 21295, 22527, 29281, 30975, 42591, 45055, 58563, 61951, 85183, 90111, 117127, 123903, 161050, 161052, 170367, 180223, 234255, 247807, 322101, 340735

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 11^t + 1, t >= 1; (b) b = 2^s*11^t - 1, s >= 0, t >= 1; (c) b = 2, 3.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 11, the nontrivial bases are 2, 3.

			1/11 has period length 10 in base 2. Note that 3, 11, 31 are the only prime factors of 2^10 - 1 = 1023, but 1/3 has period length 2 and 1/31 has period length 5, so 11 is a unique-period prime in base 2.
1/11 has period length 5 in base 3. Note that 2, 11 are the only prime factors of 3^5 - 1 = 242, but 1/2 has period length 1, so 11 is a unique-period prime in base 3.

Jianing Song, Table of n, a(n) for n = 1..590
Wikipedia, Unique prime

Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), A306075 (p=7), this sequence (p=11), A306077 (p=13).

p = 11;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A306077 Bases in which 13 is a unique-period prime.

Original entry on oeis.org

2, 3, 4, 5, 12, 14, 22, 23, 25, 51, 103, 168, 170, 207, 239, 337, 415, 675, 831, 1351, 1663, 2196, 2198, 2703, 3327, 4393, 5407, 6655, 8787, 10815, 13311, 17575, 21631, 26623, 28560, 28562, 35151, 43263, 53247, 57121, 70303, 86527, 106495, 114243, 140607, 173055, 212991, 228487, 281215, 346111

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period of the base-b expansion of its reciprocal, 1/p, is equal to the period of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 13^t + 1, t >= 1; (b) b = 2^s*13^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 22, 23, 239.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 13, the nontrivial bases are 2, 3, 4, 5, 22, 23, 239.

			1/13 has period 12 in base 2. Note that 3, 5, 7, 13, 31 are the only prime factors of 2^12 - 1 = 4095, but 1/3 has period 2, 1/5 has period 4, 1/7 has period 3, 1/31 has period 5, so 13 is a unique-period prime in base 2. (For the same reason, 13 is a unique-period prime in base 4.)
1/13 has period 3 in base 3. Note that 2, 13 are the only prime factors of 3^3 - 1 = 26, but 1/2 has period 1, so 13 is a unique-period prime in base 3.
1/13 has period 3 in base 22. Note that 3, 7, 13 are the only prime factors of 22^3 - 1 = 10647, but 1/3 and 1/7 both have period 1, so 13 is a unique-period prime in base 22.

Jianing Song, Table of n, a(n) for n = 1..554
Wikipedia, Unique prime

Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), A306075 (p=7), A306076 (p=11), this sequence (p=13).

p = 13;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A247071 Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.

Original entry on oeis.org

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, 26, 42, 13, 34, 40, 32, 54, 17, 38, 27, 19, 33, 46, 56, 90, 78, 62, 31, 80

Offset: 1

Eric Chen, Nov 16 2014

Periods associated with A144755 in base 2. The binary analog of A051627.

			2^12 - 1 = 4095 = 3 * 3 * 5 * 7 * 13, but none of 3, 5, 7 is a primitive prime factor, so the only primitive prime factor of 2^12 - 1 is 13.

Cf. A161508, A161509, A144755, A007498, A007615, A051627, A040017.

nmax = 65536; primesPeriods = Reap[Do[p = Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]]

a(n) = A002326((A144755(n+1)-1)/2). - Max Alekseyev, Feb 11 2024

Sequence trimmed to the established terms of A144755 by Max Alekseyev, Feb 11 2024

A333973 Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.

Original entry on oeis.org

18, 20, 21, 54, 147, 342, 602, 889, 258121

Offset: 1

Jeppe Stig Nielsen, Sep 22 2020

The unique prime factor of A064078(k) is then a unique prime to base 2 (see A161509), but not a cyclotomic number.
Subsequence of A161508. In fact, subsequence of the set difference A161508 \ A072226.
In all known examples, A064078(k) is a prime. If A064078(k) was a prime power p^j with j>1, then p would be both a Wieferich prime (A001220) and a unique prime to base 2.
Subsequence of A093106 (the characterization of A093106 can be useful when searching for more terms).
Should this sequence be infinite?

Henri Lifchitz and Renaud Lifchitz, Phi(258121,2)/719.
Wikipedia, Unique prime, section Binary unique primes.

Cf. A019320, A064078, A093106, A072226, A144755, A161508, A161509, A247071.

for(n=1,+oo,c=polcyclo(n,2); c % n < 2 && next(); c/=(c%n); ispseudoprime(if(ispower(c,,&b),b,c))&&print1(n, ", "))

cp's OEIS Frontend

A161508 Numbers k such that 2^k-1 has only one primitive prime factor.

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A161509 The unique primitive prime factor of 2^n-1 for the n in A161508.

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A306073 Bases in which 3 is a unique-period prime.

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A306074 Bases in which 5 is a unique-period prime.

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A306075 Bases in which 7 is a unique-period prime.

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A306076 Bases in which 11 is a unique-period prime.

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A306077 Bases in which 13 is a unique-period prime.

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A247071 Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.

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